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Theorem oawordri 7630
Description: Weak ordering property of ordinal addition. Proposition 8.7 of [TakeutiZaring] p. 59. (Contributed by NM, 7-Dec-2004.)
Assertion
Ref Expression
oawordri |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A C_ B -> ( A +o C ) C_ ( B +o C ) ) )
Proof of Theorem oawordri
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6658 . . . . 5 |- ( x = (/) -> ( A +o x ) = ( A +o (/) ) )
2 oveq2 6658 . . . . 5 |- ( x = (/) -> ( B +o x ) = ( B +o (/) ) )
31, 2sseq12d 3634 . . . 4 |- ( x = (/) -> ( ( A +o x ) C_ ( B +o x ) <-> ( A +o (/) ) C_ ( B +o (/) ) ) )
4 oveq2 6658 . . . . 5 |- ( x = y -> ( A +o x ) = ( A +o y ) )
5 oveq2 6658 . . . . 5 |- ( x = y -> ( B +o x ) = ( B +o y ) )
64, 5sseq12d 3634 . . . 4 |- ( x = y -> ( ( A +o x ) C_ ( B +o x ) <-> ( A +o y ) C_ ( B +o y ) ) )
7 oveq2 6658 . . . . 5 |- ( x = suc y -> ( A +o x ) = ( A +o suc y ) )
8 oveq2 6658 . . . . 5 |- ( x = suc y -> ( B +o x ) = ( B +o suc y ) )
97, 8sseq12d 3634 . . . 4 |- ( x = suc y -> ( ( A +o x ) C_ ( B +o x ) <-> ( A +o suc y ) C_ ( B +o suc y ) ) )
10 oveq2 6658 . . . . 5 |- ( x = C -> ( A +o x ) = ( A +o C ) )
11 oveq2 6658 . . . . 5 |- ( x = C -> ( B +o x ) = ( B +o C ) )
1210, 11sseq12d 3634 . . . 4 |- ( x = C -> ( ( A +o x ) C_ ( B +o x ) <-> ( A +o C ) C_ ( B +o C ) ) )
13 oa0 7596 . . . . . . 7 |- ( A e. On -> ( A +o (/) ) = A )
1413adantr 481 . . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( A +o (/) ) = A )
15 oa0 7596 . . . . . . 7 |- ( B e. On -> ( B +o (/) ) = B )
1615adantl 482 . . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( B +o (/) ) = B )
1714, 16sseq12d 3634 . . . . 5 |- ( ( A e. On /\ B e. On ) -> ( ( A +o (/) ) C_ ( B +o (/) ) <-> A C_ B ) )
1817biimpar 502 . . . 4 |- ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> ( A +o (/) ) C_ ( B +o (/) ) )
19 oacl 7615 . . . . . . . . . . 11 |- ( ( A e. On /\ y e. On ) -> ( A +o y ) e. On )
20 eloni 5733 . . . . . . . . . . 11 |- ( ( A +o y ) e. On -> Ord ( A +o y ) )
2119, 20syl 17 . . . . . . . . . 10 |- ( ( A e. On /\ y e. On ) -> Ord ( A +o y ) )
22 oacl 7615 . . . . . . . . . . 11 |- ( ( B e. On /\ y e. On ) -> ( B +o y ) e. On )
23 eloni 5733 . . . . . . . . . . 11 |- ( ( B +o y ) e. On -> Ord ( B +o y ) )
2422, 23syl 17 . . . . . . . . . 10 |- ( ( B e. On /\ y e. On ) -> Ord ( B +o y ) )
25 ordsucsssuc 7023 . . . . . . . . . 10 |- ( ( Ord ( A +o y ) /\ Ord ( B +o y ) ) -> ( ( A +o y ) C_ ( B +o y ) <-> suc ( A +o y ) C_ suc ( B +o y ) ) )
2621, 24, 25syl2an 494 . . . . . . . . 9 |- ( ( ( A e. On /\ y e. On ) /\ ( B e. On /\ y e. On ) ) -> ( ( A +o y ) C_ ( B +o y ) <-> suc ( A +o y ) C_ suc ( B +o y ) ) )
2726anandirs 874 . . . . . . . 8 |- ( ( ( A e. On /\ B e. On ) /\ y e. On ) -> ( ( A +o y ) C_ ( B +o y ) <-> suc ( A +o y ) C_ suc ( B +o y ) ) )
28 oasuc 7604 . . . . . . . . . 10 |- ( ( A e. On /\ y e. On ) -> ( A +o suc y ) = suc ( A +o y ) )
2928adantlr 751 . . . . . . . . 9 |- ( ( ( A e. On /\ B e. On ) /\ y e. On ) -> ( A +o suc y ) = suc ( A +o y ) )
30 oasuc 7604 . . . . . . . . . 10 |- ( ( B e. On /\ y e. On ) -> ( B +o suc y ) = suc ( B +o y ) )
3130adantll 750 . . . . . . . . 9 |- ( ( ( A e. On /\ B e. On ) /\ y e. On ) -> ( B +o suc y ) = suc ( B +o y ) )
3229, 31sseq12d 3634 . . . . . . . 8 |- ( ( ( A e. On /\ B e. On ) /\ y e. On ) -> ( ( A +o suc y ) C_ ( B +o suc y ) <-> suc ( A +o y ) C_ suc ( B +o y ) ) )
3327, 32bitr4d 271 . . . . . . 7 |- ( ( ( A e. On /\ B e. On ) /\ y e. On ) -> ( ( A +o y ) C_ ( B +o y ) <-> ( A +o suc y ) C_ ( B +o suc y ) ) )
3433biimpd 219 . . . . . 6 |- ( ( ( A e. On /\ B e. On ) /\ y e. On ) -> ( ( A +o y ) C_ ( B +o y ) -> ( A +o suc y ) C_ ( B +o suc y ) ) )
3534expcom 451 . . . . 5 |- ( y e. On -> ( ( A e. On /\ B e. On ) -> ( ( A +o y ) C_ ( B +o y ) -> ( A +o suc y ) C_ ( B +o suc y ) ) ) )
3635adantrd 484 . . . 4 |- ( y e. On -> ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> ( ( A +o y ) C_ ( B +o y ) -> ( A +o suc y ) C_ ( B +o suc y ) ) ) )
37 vex 3203 . . . . . . 7 |- x e. _V
38 ss2iun 4536 . . . . . . . 8 |- ( A. y e. x ( A +o y ) C_ ( B +o y ) -> U_ y e. x ( A +o y ) C_ U_ y e. x ( B +o y ) )
39 oalim 7612 . . . . . . . . . 10 |- ( ( A e. On /\ ( x e. _V /\ Lim x ) ) -> ( A +o x ) = U_ y e. x ( A +o y ) )
4039adantlr 751 . . . . . . . . 9 |- ( ( ( A e. On /\ B e. On ) /\ ( x e. _V /\ Lim x ) ) -> ( A +o x ) = U_ y e. x ( A +o y ) )
41 oalim 7612 . . . . . . . . . 10 |- ( ( B e. On /\ ( x e. _V /\ Lim x ) ) -> ( B +o x ) = U_ y e. x ( B +o y ) )
4241adantll 750 . . . . . . . . 9 |- ( ( ( A e. On /\ B e. On ) /\ ( x e. _V /\ Lim x ) ) -> ( B +o x ) = U_ y e. x ( B +o y ) )
4340, 42sseq12d 3634 . . . . . . . 8 |- ( ( ( A e. On /\ B e. On ) /\ ( x e. _V /\ Lim x ) ) -> ( ( A +o x ) C_ ( B +o x ) <-> U_ y e. x ( A +o y ) C_ U_ y e. x ( B +o y ) ) )
4438, 43syl5ibr 236 . . . . . . 7 |- ( ( ( A e. On /\ B e. On ) /\ ( x e. _V /\ Lim x ) ) -> ( A. y e. x ( A +o y ) C_ ( B +o y ) -> ( A +o x ) C_ ( B +o x ) ) )
4537, 44mpanr1 719 . . . . . 6 |- ( ( ( A e. On /\ B e. On ) /\ Lim x ) -> ( A. y e. x ( A +o y ) C_ ( B +o y ) -> ( A +o x ) C_ ( B +o x ) ) )
4645expcom 451 . . . . 5 |- ( Lim x -> ( ( A e. On /\ B e. On ) -> ( A. y e. x ( A +o y ) C_ ( B +o y ) -> ( A +o x ) C_ ( B +o x ) ) ) )
4746adantrd 484 . . . 4 |- ( Lim x -> ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> ( A. y e. x ( A +o y ) C_ ( B +o y ) -> ( A +o x ) C_ ( B +o x ) ) ) )
483, 6, 9, 12, 18, 36, 47tfinds3 7064 . . 3 |- ( C e. On -> ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> ( A +o C ) C_ ( B +o C ) ) )
4948exp4c 636 . 2 |- ( C e. On -> ( A e. On -> ( B e. On -> ( A C_ B -> ( A +o C ) C_ ( B +o C ) ) ) ) )
50493imp231 1258 1 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A C_ B -> ( A +o C ) C_ ( B +o C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   U_ciun 4520   Ord word 5722   Oncon0 5723   Lim wlim 5724   suc csuc 5725  (class class class)co 6650   +o coa 7557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564
This theorem is referenced by:  oaword2  7633  omwordri  7652  oaabs2  7725
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